Channel Simulation: Finite Blocklengths and Broadcast Channels

TL;DR

The reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.

Abstract

We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel's multipartite mutual information.

Figures (6)

• Figure 1: Various tasks related to remote generation of correlations.
• Figure 2: The task of (point-to-point) channel simulation.
• Figure 3: Consider i.i.d. copies of a binary symmetric channel $W$ with crossover probability $\delta = 0.1$ and an error parameter $\epsilon = 0.05$. The figure shows the second-order approximation of the common randomness assisted $\epsilon$-error simulation cost and the exact no-signaling assisted $\epsilon$-error simulation cost. This is contrasted with the corresponding quantities for channel coding, as discussed in polyanskiy2010channel, for example. The capacity of the channel is achieved for both tasks in the asymptotic limit.
• Figure 4: Asymptotic simulation region for the broadcast channel $W_{\mathsf{YZ}|\mathsf{X}} = \mathrm{BSC}_{\mathsf{Z}|\mathsf{Y}}\circ \mathrm{BSC}_{\mathsf{Y}|\mathsf{X}}$, where $\mathrm{BSC}_{\mathsf{Z}|\mathsf{Y}}$ and $\mathrm{BSC}_{\mathsf{Y}|\mathsf{X}}$ are binary symmetric channels with crossover probability $\delta = 0.3$.
• Figure 5: The task of simulating a bipartite broadcast channel using shared randomness and a pair of identity channels.

Theorems & Definitions (63)

• Lemma 1
• proof
• Theorem 2: One-Shot Achievability Bound
• proof
• Lemma 3: Classical Special Case of berta2013quantum
• proof
• Theorem 4: One-Shot Converse Bound
• proof
• Theorem 5
• Lemma 6